We assume throughout this section that \(\,\cdot \,\,\cdot \,_{\mathfrak {g}}\)  is \(\operatorname{Ad}\) -invariant, i.e. \({gXg^{-1}}{gYg^{-1}}_{\mathfrak {g}}= \\XY_{\mathfrak {g}}\)  for all \(g\in G\) , \(X,Y\in {\mathfrak {g}}\) .
An \(\operatorname{Ad}\) -invariant inner-product exists always on compact as well as abelian Lie groups [1]}.
The Riemannian geometry on \(G\)  induced by an \(\operatorname{Ad}\) -invariant inner-product on \({\mathfrak {g}}\)  is bi-invariant.
That is, the differentials of left and right-translations are isometric.
